Apart from being able to detect the plugin in all major browsers, the thing I like about the swfobject the most is the SEO benefit. Using the swfobject alows you to insert alternate content (including text) that is only seen by either people who dont have flash installed and bots (search engines), so the content will be crawled. Using this technique you can also insert a screen shot of the flash file so people with flash not installed see a still image, instead of nothing.
Another great feature of the swfobject is a thing called ExpressInstall. With this enabled it allows users with an older version of flash installed to be able to upgrade their flash player without even leaving your site, which is a really cool feature to have, especially when the swfobject is completely free.
I could give examples here of how to use the swfobject, but the official swfobject site already has great examples, and a full list of the features and benefits, plus has the latest version to download as well, so check the site out here.